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Investigation of Longitudinal Spatial Coherence for Electromagnetic Investigation of longitudinal spatial coherence for electromagnetic optical fields BHASKAR KANSERI* AND GAYTRI ARYA Experimental Quantum Interferometry and Polarization (EQUIP), Department of Physics, Indian Institute of Technology Delhi, New Delhi 110016, India. *[email protected] Abstract: For light fields, the coherence in longitudinal direction is governed by both the frequency spectra and angular spectra they possess. In this work, we develop and report a theoretical formulation to demonstrate the effect of the angular spectra of electromagnetic light fields in quantifying their longitudinal spatial coherence. The experimental results obtained by measuring the electromagnetic longitudinal spatial coherence and degree of cross-polarization of uniformly polarized light fields for different angular spectra validate the theoretical findings. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement References 1. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007). 2. V. Ryabukho, D. Lyakin, and V. Lychagov, “Longitudinal purely spatial coherence of a light field,” Opt. Spectrosc. 100, 724–733 (2006). 3. V. Ryabukho, D. Lyakin, and V. Lychagov, “Longitudinal coherence length of an optical field,” Opt. Spectrosc. 107, 282–287 (2009). 4. D. Lyakin, P. Ryabukho, and V. Ryabukho, “Mutual spatiotemporal coherence of optical fields in an amplitude- splitting interferometer,” Opt. Spectrosc. 122, 329–337 (2017). 5. I. 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Ryabukho, D. Lyakin, and M. Lobachev, “Influence of longitudinal spatial coherence on the signal of a scanning interferometer,” Opt. Lett. 29, 667–669 (2004). 12. H. Ko, “Coherence theory of radio-astronomical measurements,” IEEE Transactions on Antennas Propag. 15, 10–20 (1967). 13. J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39, 4107–4111 (2000). 14. Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express 14, 12109–12121 (2006). 15. A. Ahmad, V. Srivastava, V. Dubey, and D. Mehta, “Ultra-short longitudinal spatial coherence length of laser light arXiv:1907.05831v1 [physics.optics] 12 Jul 2019 with the combined effect of spatial, angular, and temporal diversity,” Appl. Phys. Lett. 106, 093701 (2015). 16. I. Abdulhalim, “Spatial and temporal coherence effects in interference microscopy and full-field optical coherence tomography,” Annalen der Physik 524, 787–804 (2012). 17. I. Zeylikovich, “Short coherence length produced by a spatial incoherent source applied for the Linnik-type interfer- ometer,” Appl. Opt. 47, 2171–2177 (2008). 18. J. M. Schmitt, “Optical coherence tomography (oct): a review,” IEEE J. selected topics quantum electronics 5, 1205–1215 (1999). 19. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999). 20. T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669–2671 (2006). 21. T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg, “Hanbury Brown-Twiss effect with electromagnetic waves,” Opt. Express 19, 15188–15195 (2011). 22. A. Al-Qasimi, M. Lahiri, D. Kuebel, D. F. James, and E. Wolf, “The influence of the degree of cross-polarization on the Hanbury Brown-Twiss effect,” Opt. Express 18, 17124–17129 (2010). 23. S. Volkov, D. F. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A: Pure Appl. Opt. 10, 055001 (2008). 24. L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Temporal electromagnetic degree of coherence and Stokes-parameter modulations in Michelson interferometer,” Appl. Phys. B 122, 32 (2016). 25. B. Kanseri and H. C. Kandpal, “Experimental determination of two-point Stokes parameters for a partially coherent broadband light beam,” Opt. Commun. 283, 4558–4562 (2010). 26. L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Measurement of the degree of temporal coherence of unpolarized light beams,” Photonics Res. 5, 156–161 (2017). 27. B. Kanseri, Optical Coherence and Polarization: An Experimental Outlook (Lambert Academic, 2013). 1. Introduction For optical fields, the field correlation between different points in the direction of its propagation has been mainly understood as temporal coherence [1]. However, depending on the divergence of the field, coherence in the longitudinal direction can be seen as the combined effect of the frequency spectrum and angular spectrum [2–8]. Existing studies in this direction emphasize on the role of the size of an extended source in determining longitudinal spatial coherence (LSC) [9–11], and use of interferometric schemes to determine LSC for scalar light fields [4,11]. It has been demonstrated that the angular diversity of the field results in amplitude-phase transformation and thus effective coherence length (Lc) in longitudinal direction is given in terms of temporal coherence length (lc) and longitudinal spatial coherence length (ρ ) as k 1 1 + 1 [2,3]. LSC has found several applications in applied optics [10], in radio- LC ≈ lc ρ astronomy [12]k and in surface profilometry [13,14]. In the medical field, it has been used to decrease effective coherence length of highly monochromatic source such as laser by orders of magnitude, providing high axial resolution in state of art imaging techniques such as optical coherence tomography (OCT) and interference microscopy [5,15–17]. Clearly, the role of LSC is quite important in controlling the coherence features of light sources. To extract the information of the tissue birefringence and scattering, and to study the electro- magnetic fields, the polarization of the field also needs to be examined [18]. Since polarization and coherence both characterize the statistical similarity of correlations [1,19], during the past decade or so, their combined effects have been explored in both spatial and temporal domains (for details see [20–27] and references therein). Since existing studies on LSC have mainly focussed on scalar aspects, and the role of vectorial nature of light has not been explored so far, it is quite essential and timely to investigate the combined effect of polarization with LSC both theoretically and experimentally. In this paper, first we develop a theoretical framework to quantify the effect of angular spectra on the coherence in the longitudinal direction for electromagnetic (EM) fields using the standard theory of partial coherence and polarization. Our study finds that the electromagnetic degree of coherence (EMDOC) in the longitudinal direction can be written as a product of two terms: one characterizing angular spectrum and the other temporal EMDOC of the source. Further for the validation of the theory, we conduct an experiment, in which the longitudinal EMDOC has been measured for different angular spectra of a uniformly polarized source. We also determine the degree of cross-polarization (DOCP) both theoretically and experimentally for our source of variable angular spectrum. 2. Theory Let us consider a random, statistically stationary, uniformly polarized light field propagating in z-direction emerging from a monochromatic source σ (at z = 0), as shown in Fig. 1. The extended source exhibits an angular spectrum of 2θ at point P1 0, 0, z1 in the subsequent plane and α denotes the angular coordinatedue to an arbitrarypoint s on( the source) plane. The electric Fig. 1. Schematic diagram representing the geometrical angular spectrum (2θ) of a source (σ). Symbols are described in the text. field components reaching at point Pm 0, 0, zm (m=1, 2) from an arbitrary single point s at point ( ) P1 [19] can be expressed as Ai t, α Ei zm, t, α = ( ) exp ιkrm ;for i = x, y , (1) ( ) rm ( ) ( ) where ι represents ‘iota’ (imaginary part), Ai t, α denotes the complex amplitude at point s at ′ ′ ( ) time t, rm represents the distance between points s and Pm (m=1, 2) and k is the wave vector. The coherence properties of different polarization components of light field are given by the 2 2 electric cross-spectral density (CSD) matrix [1]. CSD matrix Wij z1, z2, α = × ( ) E z1, t, α Ej z2, t, α describes the correlations between the field components at point z1 h i∗( ) ( )i and z2 due to the point source s. For the extended source σ (see sec.4.8 of [10]), this matrix is obtained by integrating the angular spectra over the source plane as 2 Wij z1, z2, θ = z E∗ z1, t, α Ej z2, t, α dζdη; (2) ( ) ¹ ¹ h i ( ) ( ) 2+ 2 = x = y = √x y s where ζ z , η z and α z and (x,y) are the coordinates of point . Using Jacobian transformation of coordinates, the integration obtained in terms of α is given by θ = 2 Wij z1, z2, θ 2πz Ei∗ z1, t, α Ej z2, t, α αdα. (3) ( ) ¹0 h ( ) ( )i Considering the general case of a uniform intensity source i.e A t, α = A t , we get the CSD matrix as ( ) ( ) πz2θ2 ιk∆z k∆zθ2 Wij zm, zn, θ = exp − sinc Wij zm, zn, 0 ; for m, n = 1, 2 , (4) ( ) 2 4 4 ( ) ( ) A t Aj t , W z z = k∆z h ∗i ( ) ( )i where (m n), and ij 1, 2, 0 exp ι z2 are the CSD matrix elements measuring pure temporal coherence.( For)∆z = 0( (single) point), Eq.
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